Q-Deformed Oscillator Algebra and an Index Theorem for the Photon Phase Operator

The quantum deformation of the oscillator algebra and its implications on the phase operator are studied from a view point of an index theorem by using an explicit matrix representation. For a positive deformation parameter q or q = exp(2πiθ) with an irrational θ, one obtains an index condition dim ker a − dim ker a † = 1 which allows only a non-hermitian phase operator with dim ker e iϕ − dim ker (e iϕ) † = 1. For q = exp(2πiθ) with a rational θ , one formally obtains the singular situation dim ker a = ∞ and dim ker a † = ∞, which allows a hermitian phase operator with dim ker e iΦ − dim ker (e iΦ) † = 0 as well as the non-hermitian one with dim ker e iϕ − dim ker (e iϕ) † = 1. Implications of this interpretation of the quantum deformation are discussed. We also show how to overcome the problem of negative norm for q = exp(2πiθ).